`intsqrt(1-4x-x^(2))dx`

Options

A. `(x)/(2)sqrt(1-4x-x^(2))+(5)/(2)sin^(-1)((x+2)/(sqrt5))+C`
B. `(x)/(2)sin^(-1)((x+2)/(sqrt5))+(5)/(2)sqrt(1-4x-x^(2))+C`
C. `(x+2)/(2)sin^(-1)((x+2)/(sqrt5))+(5)/(2)sqrt(1-4x-x^(2))+C`
D. `(x+2)/(2)sqrt(1-4x-x^(2))+(5)/(2)sin^(-1)((x+2)/(sqrt5))+C`

Answer

D

Explanation

Let `l=intsqrt(1-4x-x^(2))dx`
`=intsqrt(-(x^(2)+4x-1-2^(2)+2^(2)))dx`
`=int sqrt(-[(x+2)^(2)-(sqrt5)^(2)])dx`
`=intsqrt((sqrt5)^(2)-(x+2)^(2))dx`
`[because int sqrt(a^(2)-x^(2))dx=(x)/(2)sqrt(A^(2)-x^(2))+(a^(2))/(2)sin^(-1).(x)/(a)+C]`
`rArr " "l=(x+2)/(2)sqrt(1-4x-x^(2))+(5)/(2)sin^(-1).((x+2))/(sqrt5)+C`